Journal of Statistical Physics

, Volume 163, Issue 6, pp 1350–1393 | Cite as

On Self-Similar Solutions to a Kinetic Equation Arising in Weak Turbulence Theory for the Nonlinear Schrödinger Equation

  • A. H. M. Kierkels
  • J. J. L. Velázquez


We construct a family of self-similar solutions with fat tails to a quadratic kinetic equation. This equation describes the long time behaviour of weak solutions with finite mass to the weak turbulence equation associated to the nonlinear Schrödinger equation. The solutions that we construct have finite mass, but infinite energy. In Kierkels and Velázquez (J Stat Phys 159:668–712, 2015) self-similar solutions with finite mass and energy were constructed. Here we prove upper and lower exponential bounds on the tails of these solutions.


Self-similar solutions Fat tails Exponential tails Weak turbulence Long time asymptotics 

Mathematics Subject Classification

35C06 35B40 35D30 35Q20 45G05 



The authors acknowledge support through the CRC 1060 The mathematics of emergent effects at the University of Bonn, that is funded through the German Science Foundation (DFG).


  1. 1.
    Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York (2011)MATHGoogle Scholar
  2. 2.
    Duistermaat, J.J., Kolk, J.A.C.: Distributions: Theory and Applications. Birkhäuser, Berlin (2010)CrossRefMATHGoogle Scholar
  3. 3.
    Düring, G., Josserand, C., Rica, S.: Weak turbulence for a vibrating plate: can one hear a Kolmogorov spectrum? Phys. Rev. Lett. 97, 025503 (2006)ADSCrossRefGoogle Scholar
  4. 4.
    Dyachenko, S., Newell, A.C., Pushkarev, A., Zakharov, V.E.: Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear Schrödinger equation. Physica D 57, 96–160 (1992)ADSMathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Escobedo, M., Mischler, S., Ricard, M.R.: On self-similarity and stationary problem for fragmentation and coagulation models. Ann. Inst. H. Poincaré Anal. Non Linéaire 22, 99–125 (2005)ADSMathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Escobedo, M., Velázquez, J.J.L.: On the theory of weak turbulence for the nonlinear Schrödinger equation, Memoirs AMS 238(1124) (2015)Google Scholar
  7. 7.
    Fournier, N., Laurençot, P.: Existence of self-similar solutions to Smoluchowski’s Coagulation Equation. Commun. Math. Phys. 256, 589–609 (2005)ADSCrossRefMATHGoogle Scholar
  8. 8.
    Hasselmann, K.: On the non-linear energy transfer in a gravity-wave spectrum. Part 1. General Theory. J. Fluid Mech. 12, 481–500 (1962)ADSMathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Hasselmann, K.: On the non-linear energy transfer in a gravity-wave spectrum. Part 2. Conservation theorems, wave-particle analogy, irreversibility. J. Fluid Mech. 15, 273–281 (1963)ADSMathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Kierkels, A.H.M., Velázquez, J.J.L.: On the transfer of energy towards infinity in the theory of weak turbulence for the nonlinear Schrödinger equation. J. Stat. Phys. 159, 668–712 (2015)ADSMathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Lukacs, E.: Characteristic Functions, 2nd edn. C. Griffin & Co. Ltd., High Wycombe (1970)MATHGoogle Scholar
  12. 12.
    Menon, G., Pego, R.L.: Approach to self-similarity in Smoluchowski’s coagulation equations. Commun. Pure Appl. Math. 57, 1197–1232 (2004)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Nazarenko, S.: Wave turbulence. Lecture Notes in Physics 825. Springer, New York (2011)Google Scholar
  14. 14.
    Niethammer, B., Velázquez, J.J.L.: Self-similar solutions with fat tails for Smoluchowski’s coagulation equation with locally bounded kernels. Commun. Math. Phys. 318, 505–532 (2013)ADSCrossRefMATHGoogle Scholar
  15. 15.
    Niethammer, B., Velázquez, J.J.L.: Exponential tail behavior of self-similar solutions to Smoluchowski’s coagulation equation. Commun. PDE 39, 2314–2350 (2014)CrossRefMATHGoogle Scholar
  16. 16.
    Niethammer, B., Throm, S., Velázquez, J.J.L.: Self-similar solutions with fat tails for Smoluchowski’s coagulation equation with singular kernels. Ann. Inst. H. Poincaré Anal. Non Linéaire (2015). doi: 10.1016/j.anihpc.2015.04.002
  17. 17.
    Peierls, R.: Zur kinetischen Theorie der Wärmeleitung in Kristallen. Ann. Phys. 395, 1055–1101 (1929)CrossRefMATHGoogle Scholar
  18. 18.
    Pomeau, Y.: Asymptotic time behaviour of nonlinear classical field equations. Nonlinearity 5, 707–720 (1992)ADSMathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Simon, J.: Sobolev, Besov and Nikolskii fractional spaces: imbeddings and comparisons for vector valued spaces on an interval. Ann. Mat. Pura Appl. 157, 117–148 (1990)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland Pub. Co., New York (1978)MATHGoogle Scholar
  21. 21.
    Triebel, H.: Theory of Function Spaces. Birkhäuser, Berlin (1983)CrossRefMATHGoogle Scholar
  22. 22.
    Zakharov, V.E.: Weak-turbulence spectrum in a plasma without a magnetic field. Zh. Eksp. Teor. Fiz. 51, 688–696 (1967) [Sov. Phys. JEPT 24, 455–459 (1967)]Google Scholar
  23. 23.
    Zakharov, V.E., Filonenko, N.N.: Weak turbulence of capillary waves. Zh. Prikl. Mekh. Tekh. Fiz. 8(5), 62–67 (1967) [J. Appl. Mech. Tech. Phys. 8(5), 37–40 (1967)]Google Scholar
  24. 24.
    Zakharov, V.E., L’vov, V.S., Falkovich, G.: Kolmogorov Spectra of Turbulence I. Wave Turbulence. Springer, New York (1992)Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Institute for Applied MathematicsUniversity of BonnBonnGermany

Personalised recommendations